Let $f:\mathbb{N}\rightarrow\mathbb{R}$ such that $f(2)={3\over2}$ and $2f(n+1)=f(n)+n+1$ for all $n$. Find the points on the graph of the function $g:\mathbb{N}\rightarrow\mathbb{R}$ for which the coordinates are natural numbers, where $$g(n)=\log_2(1+f(n)).$$
Any ideas?
We know that$$2f_{n+1}-2(n+1)+2=f_n-n+1$$Let $h_n=f_n-n+1$ therefore $$2h_{n+1}=h_n$$and $$h_2=\dfrac{1}{2}$$which leads to $$h_n=\left(\dfrac{1}{2}\right)^{n-1}$$or $$f_n=\left(\dfrac{1}{2}\right)^{n-1}+n-1$$finally we have that$$g_n=\log_2(1+f_n)=\log_2(\dfrac{1+n\cdot 2^{n-1}}{2^{n-1}})=\log_2(1+n\cdot 2^{n-1})-(n-1)$$which is an integer iff $\log_2(1+n\cdot 2^{n-1})$ which means that $1+n\cdot 2^{n-1}$ should be a power of two but for $n>1$ this expression is odd and can't be a power of two and for $n=1$ we have $f_1=1$ therefore$$g_1=1$$which is an integer therefore the only $n$ to satisfy the question is $1$.